Optimal. Leaf size=31 \[ \frac{1}{4} \tanh ^{-1}\left (\sqrt{x^4+1}\right )-\frac{\sqrt{x^4+1}}{4 x^4} \]
[Out]
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Rubi [A] time = 0.0358663, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{1}{4} \tanh ^{-1}\left (\sqrt{x^4+1}\right )-\frac{\sqrt{x^4+1}}{4 x^4} \]
Antiderivative was successfully verified.
[In] Int[1/(x^5*Sqrt[1 + x^4]),x]
[Out]
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Rubi in Sympy [A] time = 3.95533, size = 24, normalized size = 0.77 \[ \frac{\operatorname{atanh}{\left (\sqrt{x^{4} + 1} \right )}}{4} - \frac{\sqrt{x^{4} + 1}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**5/(x**4+1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0384936, size = 31, normalized size = 1. \[ \frac{1}{4} \tanh ^{-1}\left (\sqrt{x^4+1}\right )-\frac{\sqrt{x^4+1}}{4 x^4} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^5*Sqrt[1 + x^4]),x]
[Out]
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Maple [A] time = 0.013, size = 24, normalized size = 0.8 \[ -{\frac{1}{4\,{x}^{4}}\sqrt{{x}^{4}+1}}+{\frac{1}{4}{\it Artanh} \left ({\frac{1}{\sqrt{{x}^{4}+1}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^5/(x^4+1)^(1/2),x)
[Out]
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Maxima [A] time = 1.43835, size = 50, normalized size = 1.61 \[ -\frac{\sqrt{x^{4} + 1}}{4 \, x^{4}} + \frac{1}{8} \, \log \left (\sqrt{x^{4} + 1} + 1\right ) - \frac{1}{8} \, \log \left (\sqrt{x^{4} + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^4 + 1)*x^5),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.260686, size = 59, normalized size = 1.9 \[ \frac{x^{4} \log \left (\sqrt{x^{4} + 1} + 1\right ) - x^{4} \log \left (\sqrt{x^{4} + 1} - 1\right ) - 2 \, \sqrt{x^{4} + 1}}{8 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^4 + 1)*x^5),x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.29972, size = 22, normalized size = 0.71 \[ \frac{\operatorname{asinh}{\left (\frac{1}{x^{2}} \right )}}{4} - \frac{\sqrt{1 + \frac{1}{x^{4}}}}{4 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**5/(x**4+1)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.227321, size = 50, normalized size = 1.61 \[ -\frac{\sqrt{x^{4} + 1}}{4 \, x^{4}} + \frac{1}{8} \,{\rm ln}\left (\sqrt{x^{4} + 1} + 1\right ) - \frac{1}{8} \,{\rm ln}\left (\sqrt{x^{4} + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^4 + 1)*x^5),x, algorithm="giac")
[Out]